At the Roots of Dictionary Compression: String Attractors
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A well-known fact in the field of lossless text compression is that high-order entropy is a weak model when the input contains long repetitions. Motivated by this fact, decades of research have generated myriads of so-called dictionary compressors: algorithms able to reduce the text's size by exploiting its repetitiveness. Lempel-Ziv 77 is probably one of the most successful and known tools of this kind, followed by straight-line programs, run-length Burrows-Wheeler transform, macro schemes, collage systems, and the compact directed acyclic word graph. In this paper, we show that these techniques are only different solutions to the same, elegant, combinatorial problem: to find a small set of positions capturing all distinct text's substrings. We call string attractor such a set. We first show reductions between dictionary compressors and string attractors. This gives us the approximation ratios of dictionary compressors with respect to the smallest string attractor and allows us to solve several open problems related to the asymptotic relations between the output sizes of different dictionary compressors. We then show that k-attractor problem --- that is, deciding whether a text has a size-t set of positions capturing all substrings of length at most k --- is NP-complete for k≥ 3. This, in particular, implies the NP-completeness of the full string attractor problem. We provide several approximation techniques for the smallest k-attractor, show that the problem is APX-complete for constant k, and give strong inapproximability results. To conclude, we provide matching lower- and upper- bounds for the random access problem on string attractors. Our data structure matching the lower bound is optimal also for LZ77, straight-line programs, collage systems, and macro schemes, and therefore essentially closes the random access problem for all these compressors.
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